Math 5593 Linear Programming Problem Set 3

Transcription

1 Math 5593 Linear Programming Problem Set 3 University of Colorado Denver, Fall 2011 Solutions (October 6, 2011) Solution 3.1 (Production/Transportation Model) [steelp.mod] set ORIG; set DEST; set PROD; # origins (steel mills) # destinations (factories) # products param rate {ORIG,PROD} > 0; param avail {ORIG} >= 0; param demand {DEST,PROD} >= 0; # tons per hour at origins # hours available at origins # tons required at destinations param make_cost {ORIG,PROD} >= 0; param trans_cost {ORIG,DEST,PROD} >= 0; # manufacturing cost/ton # shipping cost/ton var Make {ORIG,PROD} >= 0; # tons produced at origins var Trans {ORIG,DEST,PROD} >= 0; # tons shipped minimize Total_Cost: sum {i in ORIG, p in PROD} make_cost[i,p] * Make[i,p] + sum {i in ORIG, j in DEST, p in PROD} trans_cost[i,j,p] * Trans[i,j,p]; subject to Time {i in ORIG}: sum {p in PROD} (1/rate[i,p]) * Make[i,p] <= avail[i]; subject to Supply {i in ORIG, p in PROD}: sum {j in DEST} Trans[i,j,p] = Make[i,p]; subject to Demand {j in DEST, p in PROD}: sum {i in ORIG} Trans[i,j,p] = demand[j,p]; Data (Production/Transportation Model) [steelp.dat] data; set ORIG := GARY CLEV PITT ; set DEST := FRA DET LAN WIN STL FRE LAF ; set PROD := bands coils plate ; param avail := GARY 20 CLEV 15 PITT 20 ; param demand (tr): FRA DET LAN WIN STL FRE LAF := bands coils plate ; param rate (tr): GARY CLEV PITT := bands

2 Math 5593 Linear Programming Problem Set 3 (with Solutions), UC Denver, Fall coils plate ; param make_cost (tr): GARY CLEV PITT := bands coils plate ; param trans_cost := [*,*,bands]: FRA DET LAN WIN STL FRE LAF := GARY CLEV PITT [*,*,coils]: FRA DET LAN WIN STL FRE LAF := GARY CLEV PITT [*,*,plate]: FRA DET LAN WIN STL FRE LAF := GARY CLEV PITT ; AMPL Output [model steelp.mod; data steelp.dat; solve; display Make, Trans;] Presolve eliminates 1 constraint and 3 variables. Adjusted problem: 69 variables, all linear 32 constraints, all linear; 138 nonzeros 1 linear objective; 69 nonzeros. MINOS 5.5: optimal solution found. 27 iterations, objective Make := CLEV bands 0 CLEV coils 1950 CLEV plate 0 GARY bands 1125 GARY coils 1750 GARY plate 300 PITT bands 775 PITT coils 500 PITT plate 500 ; Trans [CLEV,*,*] : bands coils plate := DET FRA 0 0 0

3 Math 5593 Linear Programming Problem Set 3 (with Solutions), UC Denver, Fall FRE LAF LAN STL WIN [GARY,*,*] : bands coils plate := DET FRA FRE LAF LAN STL WIN [PITT,*,*] : bands coils plate := DET FRA FRE LAF LAN STL WIN ; Solution 3.2 (Solving LPs using the Primal-Dual and the Dual-Primal Two-Phase Simplex Methods) Solution 5.8 (Exercise 2.4): Since this problem has the immediate dual feasible solution (z 1, z 2, z 3 ) = (c 1, c 2, c 3 ) = (1, 3, 1), we can skip a primal phase-i and solve this problem directly using the regular dual simplex method (as special case of the primal-dual simplex method). In particular, after one (uniquely determined) dual pivot between x 2 and x 4, we find optimal primal and dual solutions (x 1, x 2, x 3, x 4, x 5 ) = (0, 1, 0, 0, 5) and (z 1, z 2, z 3, z 4, z 5 ) = (2.2, 0, 1.6, 0.6, 0) with the optimal objective value ζ = 3. x 1 x 2 x 3 1 ζ x x x 1 x 4 x 3 1 ζ x x Solution 5.9 (Exercise 2.6): Since the initial dictionary / tableau to this problem is both primal and dual infeasible, we need to choose one of the two two-phase methods to solve this problem. We note that both the initial primal and dual iterate have two infeasibilities (because the first two primal slacks x 4 = 3 and x 5 = 1 and the dual variables z 1 = 1 and z 2 = 3 are negative), so that it is not clear whether the primal-dual or the dual-primal simplex method would give us an easier game to play. Thus doing both, we start with the primal-dual simplex method and a first (uniquely determined) primal pivot between x 2 and x 5.

4 Math 5593 Linear Programming Problem Set 3 (with Solutions), UC Denver, Fall x 1 x 2 1 ζ x x x x 1 x 5 1 ζ x x x Since the new tableau is dual feasible, we can drop the auxiliary dual objective and continue with the regular dual simplex method. Using Bland s rule, we make a dual pivot between x 1 and x 3 x 1 x 5 1 ζ x x x x 3 x 5 1 ζ x x x resulting in an unbounded dual feasible tableau which shows that the original primal problem must be infeasible. The same conclusion follows (actually a little bit quicker) when using the dual-primal simplex method and Bland s rule that directly forces the pivot between x 1 and x 3 x 1 x 2 1 ζ x x x x 3 x 2 1 ζ x x x Solution 3.3 (Pivoting Challenge) Here are your pivot scores in increasing order for the Dual(-Phase-I)-Primal(-Phase-II) Method, the Primal(-Phase-I)-Dual(-Phase-II) Method, and your combined score fot both pivot challenges (fewer pivots is typically better, although minimum scores may also depend on some lucky choices!). Everybody who submitted scores received full credit! Dual-Primal Method Primal-Dual Method Combined Ranking Rank Score Player Rank Score Player Rank Score Player 1 13 Ganesh 1 15 Ganesh 1 28 Ganesh 2 14 YongLi 15 YongLi 2 29 YongLi 3 15 Anzhelika 3 17 Parvaneh 3 32 Parvaneh 15 Evan 4 18 Cathy 4 34 Anzhelika 15 Parvaneh 5 19 Anzhelika 34 Cathy 6 16 Cathy 19 Kapil 6 35 Evan 16 Jeremy 7 20 Evan 7 36 Kapil 8 17 Kapil 20 Lauren 8 38 Lauren 9 18 Lauren 20 Jenny 38 Jenny 18 Jenny Jeremy 38 Jeremy Solution 3.4 (Dual Formulation and Stories): Note that the given diet problem is formulated as minimization LP that has the exact same form as our regular dual problem. Hence, the associated dual problem looks like an initial primal problem m maximize b i y i subject to i=1 m a ij y i c j i=1 y i 0 j = 1, 2,..., n i = 1, 2,..., m

5 Math 5593 Linear Programming Problem Set 3 (with Solutions), UC Denver, Fall To give a proper interpretation of this problem, let us first clarify the meaning of the new dual variables as shadow prices for the original primal constraints. To do that, we may look at the units [y] = [c]/[a] = ($/unit food)/(nutrient content/unit food) = $/nutrient content which suggests that the y i correspond to the shadow prices for each of the m nutrients i = 1,..., m. In particular, then the objective is still to maximize a dollar rather than a nutrient amount, because [b] [y] = (nutrient amount) ($/nutrient content) = $ Hence, the dual problem tries to price each nutrient (possibly by some food provider) as to maximize overall selling price (profit) while not exceeding current market prices for the food items that include these nutrients. Alternatively, you may remember that the dual variables contain some information on how the objective will change if we change the original constraints right-hand sides (by some sufficiently small amount, otherwise we may have to change the optimal basis): if nutrient requirement b i increases by 1 unit, then our diet cost increases by y i dollars. Hence, as before we would interpret y i as the shadow price for nutrient b i, and the story could continue as before. For your enjoyment and comparison to some other ideas, here are some of your own stories in loosely decreasing order of truthfulness to the actual situation (many are fully correct, so the first are ordered by some sort of creativity ). One point each was deducted if you maximized nutrients or nutritional requirements, or interpreted the constraint as to stay within some budget (note that c is not a budget, but the food prices or costs). Matt wants to set the prices for the nutrients so he can make a profit. The minimum daily requirements for each nutrient is predetermined, presumably by corrupt officials at the USDA who surely receive a kick-backs from Sam and his cohorts in the powerful nutrient lobby. Sam s profit then is directly related to the price that he and his cohorts (ultimately, this is a vast, nearly undetectable price-setting scheme) determines for each nutrient. The constraints proceed from the fact that the price for each of several basic food items is also predetermined (by the equally powerful and corrupt food item lobby). Therefore, the sum of the amount of each nutrient in a specific food item (the conspiracy deepens) times the price for each nutrient cannot exceed the price of that food item. Suppose a company wants to make a super pill, to market to graduate students so that the students don t have to waste time eating they can get all their day s nutrients in one convenient form without having to stop studying. They talk to the student who determined his daily nutrients and food prices, so they have the same data (assuming this student is typical of all graduate students). The company determines what price to sell the pills for by looking at price (to charge) per unit of each required nutrient, which we will call y i. Since the people this pill is marketed for are poor, in addition to busy, they won t pay for pills that satisfy more than the minimum daily requirements under this pricing model. So the company plans to charge m i=1 b iy i for each pill. They then want to maximize this amount, in order to make the most money (assuming that production costs are fairly constant and inexpensive across all nutrients). However, they know that no student is going to pay more for this pill than they would for the frozen dinners and pizza that they already determined they could get for a certain minimal price (the objective function of the primal problem). So, their prices must also satisfy the constraint that m i=1 a ijy i c j, for each food product (j = 1,..., n). And of course, they want to charge a nonnegative amount of money per nutrient, so they need y i 0. Let s imagine that we are planning to open a market near MIT and to price the food so that profit is maximized, in the sense that students prefer the new food items over the current food on the student s list. What we should do is to price each nutrient per unit first and then price a food item based on the unit prices and the units of nutrients that the food item contains.

6 Math 5593 Linear Programming Problem Set 3 (with Solutions), UC Denver, Fall More preciously, the objective is to maximize the profit, which equals the necessary daily requirements that the student needs (the student can get this information from the library) times the unit prices we plan to put on. These prices are under the constraints that the sum of each unit nutrient in corresponding food times the unit prices must be less than to the prices on the student s list, so that these constraints make sure that student prefer the new food because of its cheaper price and the satisfaction of the nutritional requirements. Note that if some constraint j is not satisfied, then the student can buy food item j to some extent instead of entirely buying from us to meet the need of nutrients with less money spent. In other words, we have food item j that is priced higher than in the market, these m constraints are posed to insure that students have no motivation to buy any food instead of just buying our food. This could be what the grocery store will be selling their vitamin/mineral/dietary supplements for. They want to offer these supplements, but they wouldn t sell the supplements for more than what a customer could get by purchasing these nutrients contained in foods, otherwise the foods would be bought rather than the supplements, yet they still want to make as much money as possible on their sales. In the above story, I want to add another person (namely a wholesaler). Suppose that the student went to the wholesaler with his list of different food items and nutritional requirements. When the wholesaler got this list, he decided to provide a diet to the student that meets his minimum daily requirements. On the other hand, the wholesaler wanted to maximize his revenue. The wholesaler thought how could he set the prices y j per unit of nutrient a ij in the food item x j so that the student would buy the food items. The student could buy his food item only if the total cost for food item x j is below c j. This restriction imposes the constraints. His total revenue guaranteed from the nutrients sale forms the objective. His total revenue guaranteed from the nutrients sale can be found by solving the dual problem. There are some producers who produce the n kinds of food which the MIT student needs. The n kinds of food are made from m different nutrients. So there is a wholesaler who supplies the producers with the m nutrients needed to make n kinds of food. The producers who produce the j (j = 1, 2,..., n) informs the supplier (wholesaler) that food j contains i (i = 1, 2,... m) nutrients and he intends to purchase the number of nutrients i in food j (a ij ) to meet the MITs student s minimum nutritional requirements b i. The supplier now has an optimization problem as follows: How can I set the each price of nutrient i (y i ) so that the producer will buy from me, and so that I will maximize my income m i=1 y ib i? The supplier also thinks over that producer j will buy only if the total cost of nutrient i for food j is below the price c j ; otherwise he runs the risk of making a loss if the MIT student opts to buy food j. This restriction imposes the constraint m i=1 a ijy i c j on the prices of the food. A vitamin manufacturer wants to maximize the cost/price b T y of his multi-vitamin. But he wants to guarantee that the cost of his multi-vitamin is competitive with the cost of food that provides the same amount of each nutrient, A T y c. A scenario in which the dual would be a natural problem would be for a person (perhaps an athlete, or someone else on a restricted diet) who wants to maximize their nutrient intake while not going over the cost of what it would have normally been to buy food to attain those nutrient levels. Perhaps they are buying supplements (because they can avoid calories and fats) and want to get nutritional intake but don t want their cost to exceed what would have been the cost had they bought the food instead. Mr. MIT really lived in this fantasy world where he would come into the grocery store and spend a small amount on his necessary groceries. On the other side of things however, Mr. KingSooper Jr. s wife was expecting a baby and they were starting to plan how to best

7 Math 5593 Linear Programming Problem Set 3 (with Solutions), UC Denver, Fall allocate their money. Living on a strict budget was hard but they figured they could raise some of the prices on things sold within the family market. Raising prices too much would of course cause their customers to turn to Mr. Safeway s store, and put Mrs. KingSooper in early labor. Both of these would be incredibly risky, so instead Mr. KingSooper registered for Alex s LP class for next semester so that he could figure out how much was the right amount to charge so that he maximizes the amount he makes all while pleasing hungry Mr. MIT. Like most Whole Food shoppers, Mr. KingSooper Jr. must keep in mind that many of his customers are health conscious out in CO. Another person in the story who would naturally want to solve the dual problem would be the owner of the grocery store from where the student shops for his food. Here y i represents the price for the nutrients. The dual problem ensures the owner the maximum revenue he can earn by selling the food to the student while satisfying the student s minimum dietary requirement and his budget. Let s assume that King Supers also delivers pizza. Then this would be the LP that the manager of King Supers would like to solve in order to maximize profit, subject to the constraints that his costumer still gets his minimum daily nutritional intake on his budgeted small stipend. The dual of the problem can be described as the roommate. His objective is to maximize the amount of nutrients he can receive from his food basket items. However, he is restricted by the price of each food item. He cannot take an amount of nutrients from a food item greater than the amount those nutrients cost compared to one unit of that food item. The dual would be to maximize the minimum daily requirements, subject to the cost of the food being less than or equal to his budget. His mother or doctor would naturally want him to solve the dual. The MIT student was very excited about his diet plan and called his mother to tell her the good news. Unfortunately, his mother was not quite as impressed. While she appreciated the mathematical thinking behind the problem, she did not like that he was focusing on spending the least amount of money while getting the minimum number of nutrients. Being his mother (and a math lover), she suggested he alter his linear problem to focus on maximizing his nutrients while staying within his budget, and then taught him all about the dual format! So, his new problem does just that maximize the number of nutrients he gets while keeping the total cost of all his food within budget. The dual should represent maximized nutrition. If your story is missing, I was probably missing your L A TEX source on Blackboard. Of course, you can still send it to me by and I will add it to this growing collection. Also, if any of the above solutions gives you a new idea, feel free to write it up and I d love to hear about it! Please let me know if you have any questions, comments, corrections, or remarks!